Abschätzungen von Kähler-Differenten
Dissertation, Universität des Saarlandes, 1999
Complete text as PDF-Document (German; 568 kByte)

Abstract

Kähler differents were introduced by Erich Kähler in his work "Algebra und Differentialrechnung" (1953). There he defined "differents" as the Fitting ideals of the ring extension's differential module.

Kähler related these differents to the ramification index of the corresponding ring extension generalizing a well-known result in algebraic number theory. The advantage of Kähler differents in relation to the more traditional "Dedekind differents" is that they can be defined under less restrictive assumptions (the residue class field extension does not need to be separable).

As is the case with Dedekind differents, one can relate the different to the ramification index of the ring extension; here the inseparability of the residue class field extension must be considered as well. Contrary to the separable case, the inseparable case allows only for an estimate of the Kähler different against ramification index and inseparability.

In a later work (Geometria Aritmetica, 1958) Kähler generalized his results once more: he replaced the differential module of a ring extension (which is not alway finitely generated, and so does not always allow for the definition of Fitting ideals) with the universally finite differential module.

Later work on Kähler differents was done by F.K. Schmidt (unpublished), whose work is referenced in a paper by Hans Hirt ("Der Differentialmodul eines lokalen Prinzipalringes über einem beliebigen Ring", 1967). There are some interesting points about Hirt's paper

• The paper considers only the situation of a ring extension S/R where S is a principal ideal ring
• Hirt does not give an estimate for Kähler differents; instead he gives estimates for the elementary divisors of the differential module.
• Almost all proofs are invalid, since they rely on one crucial lemma, which is wrong. Interestingly, most results are correct all the same. The estimates of Hirt are a special case of the results of my thesis.

Additional work on Kähler differents was done by Zhaohua Luo ("On the Ramification Theory of Regular Schemes", 1994 and "Formal Kähler Different and Uniform Bound for Morphisms of Regular Local Rings", 1995). Luo's results are special cases of estimates that were given by Kähler. However Luo gives another type of proof, which is based on the behaviour of Kähler differents under quadratic transformations. Since Kähler differents in general "do not behave well" in a tower of ring extensions, Luo needs a number of additional assumptions on the ring extension and can only give an estimate for the zeroth Kähler different.

In my thesis estimates for the elementary divisors of the differential module and for the Kähler differents of a ring extension are given, from which all the preceding results can be derived. In the case of an inseparable residue class field extension, the results improve upon the results of Kähler and Luo.